Research Papers

Nonlinear Vibration of Orthotropic Rectangular Membrane Structures Including Modal Coupling

[+] Author and Article Information
Dong Li

School of Civil Engineering,
University of Chongqing,
83 Shabei Street,
Chongqing 400045, China;
Department of Structural Engineering,
University of California San Diego,
9500 Gilman Drive 0085,
La Jolla, CA 92093-0085
e-mail: 20151601001@cqu.edu.cn

Zhou Lian Zheng

School of Civil Engineering,
University of Chongqing,
83 Shabei Street,
Chongqing 400045, China;
Chongqing Jianzhu College,
Chongqing 400072, China
e-mail: zhengzl@cqu.edu.cn

Michael D. Todd

Department of Structural Engineering,
University of California San Diego,
9500 Gilman Drive 0085,
La Jolla, CA 92093-0085
e-mail: mdtodd@ucsd.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 14, 2018; final manuscript received March 9, 2018; published online March 30, 2018. Assoc. Editor: Pedro Reis.

J. Appl. Mech 85(6), 061004 (Mar 30, 2018) (9 pages) Paper No: JAM-18-1032; doi: 10.1115/1.4039620 History: Received January 14, 2018; Revised March 09, 2018

The membrane structure has been applied throughout different fields such as civil engineering, biology, and aeronautics, among others. In many applications, large deflections negate linearizing assumptions, and linear modes begin to interact due to the nonlinearity. This paper considers the coupling effect between vibration modes and develops the theoretical analysis of the free vibration problem for orthotropic rectangular membrane structures. Von Kármán theory is applied to model the nonlinear dynamics of these membrane structures with sufficiently large deformation. The transverse displacement fields are expanded with both symmetric and asymmetric modes, and the stress function form is built with these coupled modes. Then, a reduced model with a set of coupled equations may be obtained by the Galerkin technique, which is then solved numerically by the fourth-order Runge–Kutta method. The model is validated by means of an experimental study. The proposed model can be used to quantitatively predict the softening behavior of amplitude–frequency, confirm the asymmetric characters of mode space distribution, and reveal the influence of various geometric and material parameters on the nonlinear dynamics.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Chadha, M. , and Todd, M. D. , 2017, “ A Generalized Approach for Reconstructing the Three-Dimensional Shape of Slender Structures Including the Effects of Curvature, Shear, Torsion, and Elongation,” ASME J. Appl. Mech., 84(4), p. 041003. [CrossRef]
Kumar, N. , and DasGupta, A. , 2017, “ On the Static and Dynamic Contact Problem of an Inflated Spherical Viscoelastic Membrane,” ASME J. Appl. Mech., 82(12), p. 121010.
Liu, C. J. , Deng, X. W. , and Zheng, Z. L. , 2017, “ Nonlinear Wind-Induced Aerodynamic Stability of Orthotropic Saddle Membrane Structures,” J. Wind Eng. Ind. Aerodyn., 164, pp. 119–127. [CrossRef]
Hu, Y. , Chen, W. J. , Chen, Y. F. , Zhang, D. X. , and Qiu, Z. Y. , 2017, “ Modal Behaviors and Influencing Factors Analysis of Inflated Membrane Structures,” Eng. Struct., 132, pp. 413–427. [CrossRef]
Kang, S. W. , and Lee, J. M. , 2002, “ Free Vibration Analysis of Composite Rectangular Membranes With an Oblique Interface,” J. Sound Vib., 251(3), pp. 505–517. [CrossRef]
Houmat, A. , 2005, “ Free Vibration Analysis of Membranes Using the h-p Version of the Finite Element Method,” J. Sound Vib., 282(1–2), pp. 401–410. [CrossRef]
Wu, W. X. , Shu, C. , and Wang, C. M. , 2007, “ Vibration Analysis of Arbitrarily Shaped Membranes Using Local Radial Basis of Function-Based Differential Quadrature Method,” J. Sound Vib., 306(1–2), pp. 252–270. [CrossRef]
Amore, P. , 2009, “ A New Method for Studying the Vibration of Non-Homogeneous Membranes,” J. Sound Vib., 321(1–2), pp. 104–114. [CrossRef]
Noga, S. , 2010, “ Free Transverse Vibration Analysis of an Elastically Connected Annular and Circular Double-Membrane Compound System,” J. Sound Vib., 329(9), pp. 1507–1522. [CrossRef]
Soares, M. R. , and Goncalves, P. B. , 2014, “ Large-Amplitude Nonlinear Vibrations of a Mooney–Rivlin Rectangular Membrane,” J. Sound Vib., 333(13), pp. 2920–2935. [CrossRef]
Zheng, Z. L. , Liu, C. J. , He, X. T. , and Chen, S. L. , 2009, “ Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection,” Math. Prol. Eng., 2009, p. 634362.
Liu, C. J. , Zheng, Z. L. , Long, J. , Guo, J. J. , and Wu, K. , 2013, “ Dynamic Analysis for Nonlinear Vibration of Prestressed Orthotropic Membranes With Viscous Damping,” Int. J. Struct. Stab. Dyn., 13(02), p. 1350018. [CrossRef]
Liu, C. J. , Todd, M. D. , Zheng, Z. L. , and Wu, Y. Y. , 2018, “ A Nondestructive Method for the Pretension Detection in Membrane Structures Based on Nonlinear Vibration Response to Impact,” Struct. Health Monit., 17(1), pp. 67–79. [CrossRef]
Guo, J. J. , Zheng, Z. L. , and Wu, S. , 2015, “ An Impact Vibration Experimental Research on the Pretension Rectangular Membrane Structure,” Adv. Mater. Sci. Eng., 2015, p. 387153.
Zheng, Z. L. , Lu, F. M. , He, X. T. , Sun, J. Y. , Xie, C. X. , and He, C. , 2016, “ Large Displacement Analysis of Rectangular Orthotropic Membranes Under Stochastic Impact Loading,” Int. J. Struct. Stab. Dyn., 16(01), p. 1640007. [CrossRef]
Li, D. , Zheng, Z. L. , Liu, C. Y. , Zhang, G. X. , Lian, Y. S. , Tian, Y. , Xiao, Y. , and Xie, X. M. , 2017, “ Dynamic Response of Rectangular Prestressed Membrane Subjected to Uniform Impact Load,” Arch. Civ. Mech. Eng., 17(3), pp. 586–598. [CrossRef]
Laura, P. A. A. , Bambill, D. V. , and Gutierrez, R. H. , 1997, “ A Note on Transverse Vibration of Circular, Annular, Composite Membranes,” J. Sound Vib., 205(5), pp. 692–697. [CrossRef]
Touze, C. , Thomas, O. , and Huberdeau, A. , 2004, “ Asymptotic Non-Linear Normal Modes for Large-Amplitude Vibrations of Continuous Structures,” Comput. Struct., 82(31–32), pp. 2671–2682. [CrossRef]
Lau, S. , and Cheung, Y. K. , 1981, “ Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems,” ASME J. Appl. Mech., 48(4), pp. 959–964. [CrossRef]
Goncalves, P. B. , and Del Prado, Z. J. G. N. , 2002, “ Nonlinear Oscillations and Stability of Parametrically Excited Cylindrical Shells,” Meccanica, 37(6), pp. 569–597. [CrossRef]
Goncalves, P. B. , Silva, F. M. A. , and Del Prado, Z. J. G. N. , 2008, “ Low-Dimensional Models for the Nonlinear Vibration Analysis of Cylindrical Shells Based on a Perturbation Procedure and Proper Orthogonal Decomposition,” J. Sound Vib., 315(3), pp. 641–663. [CrossRef]
Lazarus, A. , Thomas, O. , and Deü, J. F. , 2012, “ Finite Element Reduced Order Models for Nonlinear Vibrations of Piezoelectric Layered Beams With Applications to NEMS,” Finite Elements Anal. Des., 49(1), pp. 35–51. [CrossRef]
Pesheck, E. , Pierre, C. , and Shaw, S. W. , 2002, “ Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes,” ASME J. Vib. Acoust., 124(2), pp. 229–236. [CrossRef]
Kerschen, G. , Peeters, M. , Golinval, J. C. , and Vakakis, A. F. , 2009, “ Nonlinear Normal Modes, Part I: A Useful Framework for the Structural Dynamicist,” Mech. Syst. Signal Process., 23(1), pp. 170–194. [CrossRef]


Grahic Jump Location
Fig. 1

Model of rectangular orthotropic membrane structure with pretension

Grahic Jump Location
Fig. 3

Dimensionless amplitude w/h versus dimensionless first-order frequency Ω/ω11. The circular points represent experimental results abstracted from displacement–time signals. The dashed lines are predictions based on assumed stress functions with different modal orders.

Grahic Jump Location
Fig. 10

Amplitude reduction rate (A−A0)/A, versus dimensionless pretension force F/F0

Grahic Jump Location
Fig. 2

Mechanical experiment: (a) photo of experiment and (b) distribution of measured points on membrane surface

Grahic Jump Location
Fig. 4

Dimensionless displacement w/h versus dimensionless time t¯: (a) point O and (b) point B1. The membrane material is XYD brand, and initial displacement w¯0 is 2.1. The solid lines represent measured data in experiment. The dashed lines are the corresponding theoretical prediction.

Grahic Jump Location
Fig. 5

Dimensionless amplitude w/h versus dimensionless coordinate position x/a on a one-quarter membrane surface at t¯=0. The scattered points represent measured data along the weft direction in experiment. The continuous lines are predictions from theory.

Grahic Jump Location
Fig. 6

Mode shapes with different initial amplitude: (a) the first-order mode shape and (b) the second-order mode shape. The solid lines represent the mode shape based on the analysis including modal coupling. The dashed lines represent the mode shape based on the analysis without modal coupling.

Grahic Jump Location
Fig. 7

Evolution of the mode shapes throughout a quarter-period of transverse motion: (a) evolution of the first-order mode shape and (b) evolution of the second-order mode shape

Grahic Jump Location
Fig. 8

Dimensionless amplitude w/h versus dimensionless first-order frequency Ω/w11, for different values of membrane material thickness (h = 0.8 mm, 1.0 mm, 1.2 mm)

Grahic Jump Location
Fig. 9

Dimensionless first-order frequency Ω/w11 versus dimensionless pretension force F/F0 with different values of membrane aspect ratio (κ=1, 2, 3)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In